# Tilings (Math 285)

Instructor: Igor Pak, MS 6125, pak@math.

Class Schedule: MWF 12:00-12:50, MS 7608.

## Brief outline

We will give an introduction to the subject, covering a large number of classical and a few recent results. The emphasis will be on the main ideas and techniques rather than proving the most recent results in the field. The idea is to to give a guided tour over (large part of) the field to prepare for more advanced results in the future.

## Content:

I will be posting most of the literature I will be covering, numbering them roughly in order of the lectures (some lectures are collapsed into one item). A few of these papers will not be taught - they are included the source of additional reading closely related to lecture material.
1. Domino tilings
2. Combinatorial group theory
3. Ribbon tilings of Young diagram shapes
4. Tile invariants and ribbon tilings of general regions
5. Rectangles with one side integral
6. Tilings with two bars
7. T-tetromino tilings
8. Further applications of height functions
9. Tilings of rectangles
10. Tilings of rectangles with rectangles
11. Order of tiles
12. Augmentability
• Yang's paper (see above); hardness of augmentability and decidability of augmentability with rectangles.
• Korn's thesis (see above), Section 11; negative solution of augmentability for dominoes.
13. Valuations
14. Counting domino tilings and perfect matchings
• L. Lovász, M.D. Plummer, Matching Theory, Chapter 8; among other things, general Pfaffian-Determinant approach.
• R. Kenyon, An introduction to the dimer model (2002); variations on Kasteleyn method, isoradial graphs.
• R.W. Kenyon, J.G. Propp, D.B. Wilson, Trees and Matchings (2000); Temperley's bijection and applications.
15. Aztec diamond

Warnings: Most of these links are external, some are by subscription, some can be broken; occasionally, their content is unverified. Also, the explanations are NOT review, but rather quick summary of material I used from the sources; often there is wealth of other work presented there as well.

## General references:

• Federico Ardila and Richard P. Stanley, Tilings (2005), an introductory article.
• Branko Grunbaum and G. C. Shephard, Tilings and Patterns (1986), an instant classic. A generic introduction to the subject.
• Solomon W. Golomb, Polyominoes: Puzzles, Patterns, Problems, and Packings, true classic in the recreational literature (Second Ed., 1995).